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The Mathematical Foundations of Economic TheoryAuthor(s): R. G. D. AllenSource: The Quarterly Journal of Economics, Vol. 63, No. 1 (Feb., 1949), pp. 111-127Published by: Oxford University PressStable URL: http://www.jstor.org/stable/1882736.

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THE MATHEMATICAL FOUNDATIONS

OF ECONOMIC

THEORY

SUMMARY

I. General comments: the use of mathematics by Hicks and by Samuelson,

111.-

II. Samuelson's treatment of cost and

production,

114.- III.

Hicks

and Samuelson

on

consumers'

demand,

118.- IV.

Samuelson's dynamics:

difference

equations,

121.

There is

no

longer any doubt

that

mathematical

methods are

appropriately

and

usefully employed

in

the

development

of

economic

theory.

The

question, rather,

is whether the

mathematics should be

discarded

in

the final

exposition

or whether they should take their

place in the main argument. Do mathematics

form the scaffolding

or the

steel

framework of the

structure?

Marshall used mathematical methods

-

relatively simple ones

as

a

scaffolding

to assist

him

in

constructing

his

theory; they were

discarded

when he

had finished.

The

Principles

suffer,

I

believe,

from

this

fact;

if

we had been allowed

to see more of

the mathematical

reasoning, we would have found fewer points

of ambiguity and a

generally tighter exposition.

Be this as

it

may.

The main

points

are that Marshall and

many

of

his

contemporaries

were content

with

quite simple

mathematical

arguments

and

that

the use

of

mathe-

matics

in

economics

has

since

developed

both

in

scope

and

in

com-

plexity. The ways

in

which mathematics are used

by many theorists

are such that

they

cannot

be

discarded without

leaving

the

argument

defective

and full

of

expressions

such

as it can

be

proved

that ....

It is still

possible, however,

to confine the

mathematical

develop-

ment to

appendices.

The

completed

structure

can

be

described

in

general

terms

and, when the

details of

the construction need to

be

shown

-

when it

is important

to know

that the structure has

a

steel

frame

-

then

reference can

be

made to technical

appendices.

This is

the method

adopted by

J. R.

Hicks, particularly

in

Value and

Capital,

which has appeared

in

a

second edition incorporating important

revisions and extensions.' It

is, undoubtedly,

a

successful method

in

the hands of

a

master craftsman

like Hicks.

The other possibility

is the

incorporation

of mathematics into the

1. J. R.

Hicks:

Value and

Capital

(2d Edition,

Oxford University Press,

1946).

ill

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112

QUARTERLY

JOURNAL

OF

ECONOMICS

main text,

and

this method

is well

illustrated

by

P. A. Samuelson,

in his

recently

published

Foundations

of

Economic

Analysis.2

Here

the structure

is

built,

stage by stage,

from

the foundations

up.

If the

most difficult

mathematics

come

early

on,

in the steel

framework

of

the building,

that's

the

way it is.

The embellishments

are not

there

to

be admired until

the frame

is up.

Samuelson's

book

may

have

less popular

appeal

than Hicks'

but

it is, none

the

less, of great

importance.

Every

economic

theorist

worthy

of the

name will

make

a serious

effort

to examine

it

closely.

The two

books

of Hicks

and

Samuelson

have

much

the

same

subject matter, the foundations

of economic

theory;

they

follow

much

the

same

principles,

which I

regard

as undoubtedly

the correct

ones.

They

must

be studied together,

particularly

in

the

light

of the

his-

torical

development

of their

respective

theses.

Since 1937

or 1938,

each author

has been

re-shaping

and

rounding

off

his theory

-- which

had taken

a fairly

comprehensive

form before

the

war

-

and each

has been

influenced

(though,

unfortunately,

rather

remotely

influ-

enced

in the

geographical

sense)

by the

work of

the other.

They

have now come together on essentials. Future development, I

believe,

will not be Hicksian

or Samuelsonian

but will flow from

an

agreed

combination

of the

two

expositions.

Postgraduate

courses

and seminars

in

economic theory

will be

concerned

with

this

develop-

ment for

years

to

come.

Hicks

and

Samuelson,

however,

had

different

objects

in mind in

writing their

books.

Hicks

tries

to work out,

if

not a complete

economic theory,

at

least a

full

development

of

one

particular

line of

approach. He is not concerned if some of his conclusions can be

reached by

other

and

perhaps

better

roads.

Samuelson's

object is

to

unify

diverse

fields

of

economic theory

by

showing up

the

common,

underlying

mathematical

basis.

He

is

most

concerned

with

how

conclusions

are

reached,

with

what is valid

and what is false

in

diverse

theories.

Samuelson

concentrates

attention

on

operational

or

meaningful

conclusions, i.e.,

conclusions

which

could,

at least under

ideal condi-

tions, be validated or refuted by empirical data. His main unifying

principle

is

that such

conclusions

are to be derived,

not from the

equations

of

equilibrium,

but

from

the

inequalities

which

ensure

a

maximum

(minimum)

position

or which

are required

for

stability.

Another

point

he

makes

is that

it is

just

as

easy,

and sometimes

easier,

to

handle

simultaneous change

in

many

variables

as

in

a few. The

achievement

of

Walras

and

Pareto

was to

show the

essential

sim-

2.

P. A. Samuelson:

Foundationsof

Economic

Analysis

(Harvard

University

Press, 1947).


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FOUNDATIONS

OF ECONOMIC

THEORY

113

plicity of equilibrium

of many variables.

Samuelson's

mathematical

technique goes

beyond

Walras'

and Pareto's

and

deals easily with

many variable

changes

or shifts,

something

which

is rather lost in

Hicks'

non-mathematical

exposition

of comparative

statics.3 A third

point is that

results

can be

obtained, and

often easily

obtained, in

terms

of

finite changes

and

discontinuity, and

not only

in

differential

or

continuous

forms. Indeed,

if

conclusions

are to

measure

up to

empirical

data,

this

is almost essential.

Like

Hicks, then,

Samuelson

is much concerned

with comparative

statics,

with

the answers

to

such questions

as: if demand

shifts

upwards, does the price increase? He goes

on,

moreover,

to a sketch

of

what might

be

achieved

in

a field

about

which

very

little has been

said,

comparative

dynamics.

It is often thought

that

we

progress

naturally from

statics

to comparative

statics

and that

we then switch

into

something different

called

dynamics,

something

better

and more

realistic.

Samuelson

emphasizes

that

statics and

dynamics

refer to

the formulations

of economic

systems.

Moreover,

all

formulations

which involve

time are

not dynamic;

a static

system

can easily include

a secular or historical movement. Finally, Samuelson maintains,

statics and dynamics

cannot

be

kept

as

quite

distinct branches

of

analysis.

Comparative

statics

can

be

defined as

the

comparison

of

one position

of

equilibrium

with

another without reference

to the

path of transition

from

one

to the

other.

This does not

appear

to

involve

any dynamic

formulation.

But Samuelson

shows

that

meaningful

conclusions

in comparative

statics

come

from

conditions

of

stability

of

equilibrium

positions.

And

these

can

only

be

derived

from a dynamic model which shows under what circumstances a

displacement

from

equilibrium

will

be

followed

by

a return

(perhaps

oscillatory)

to equilibrium.

It

is

just

not possible

in

any

one review

to

deal

with

all

aspects

of Samuelson's book.

In

the following

sections,

I concentrate

on

certain basic

matters

which

happen

to interest

me.

I

would

add

that

what

makes

the book such essential

reading

for the economist

is not

so much Samuelson's

mathematical

treatment

as his economic

insight. So many things which have worried the economist for years

past

appear

so

simple

in

Samuelson's devastating

analysis.

He

is

quite

ruthless

in

casting

out

what is

really

irrelevant.4

3.

Hicks'

exposition

is

made

possible

by

the

use

of

the

concept

of a com-

posite

commodity on

the

assumption

that

the

prices

of

the

components

vary

in

proportion.

One

result

is

that

he

gets

rather

tangled

up

with

various

concepts

of

money.

4.

This

may

be

the

place

to

register

some

(relatively

minor)

complaints.

Samuelson

has

clearly

been lax

in

editing

and

proofreading

his

book;

there

are

numerous misprints and slips, some quite confusing, and the system of cross


8/11/2019 1882736

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114 QUARTERLY

JOURNAL OF ECONOMICS

II

The most definitive parts

of Samuelson's work are those

on

the

theory of cost and production and on the theory of consumers'

demand. Here he has

undoubted success

in

his attempt to unify,

simplify

and

codify existing theories. Many

of

the constructions

and concepts which have exercised the minds of economists

in

the

past are

shown to

be

of

little

significance, except perhaps

for

exposi-

tory purposes. The linear

homogeneous production function,

con-

sumers' surplus,

and

the

assumption

of constant

marginal utility

of

money are only some of the

more notable casualties. Some may

wish to revive them, but very potent restoratives will be needed.

Samuelson himself has very little use

for

them;

he

remains somewhat

uncertain

only

in

the case

of

the integrability conditions

on

the

consumers' scale

of

preferences, which (as

an

economist)

he

would

like to dismiss but which (as a mathematician) he cannot quite ignore.

Samuelson differs from

Hicks

in

his

treatment

of the

theory

of

production.

He takes

it before, and

not

after, the theory

of con-

sumers' demand. He presents it

in a

stage by stage analysis,

in

contrast to the wider but more formal method of Hicks. There is

room for both treatments

although, personally,

I

prefer Samuelson's.

His analysis throws more

light on matters which have troubled

economists and which have been

the subject

of heated

controversy.

The implications of pure

competition, the adding up problem

and

the

question

of

discontinuities

in

the production function are

examples. Samuelson, however,

assumes that the

firm

has

only

one

product;

he

might well have indicated the obvious extensions to the

case of joint production.

The

stages

in

the

analysis are: (1) the

combination of

inputs

(at given prices

to

the firm)

to

minimize cost

for a

given output;

(2) the choice of output to maximize net revenue to the firm; and (3)

the

external

relations

of a

firm

to

the rest

of the

industry.

The

first

references is inadquate. He is

sometimes obscure in his wording and he might

well have

spared

us such

monstrous concoctions

as

monotonicity (p. 12).

He

is not

always happy on the

mathematical side and tends to

fall between two stools.

His mathematical treatment is not simplified enough for the economist and not

rigorous enough to satisfy the

mathematician. (As a small example,

on

pp.

65-

66,

he

says: This can be proved

rigorously

in

two

ways

...

. .

But when

he

comes to the

second proof he starts:

More rigorously

.

.

. ).

His

compromise

is

particularly unsatisfactory

in

his

handling

of

matrix algebra.

He would

seem

to

have decided

to keep

the

matrix notation to footnotes, without

detailed

explani-

ations; unfortunately,

matrices

have

tended to

creep

back

into the text

ili

a

rather untidy and confusing way.

A text on matrix algebra suitable

for

the

economist is

badly

needed and it is a

pity

that

Samuelson

has not added

a third

mathematical

appendix

to

those

in

which he discusses quadratic

forms and

difference

equations.


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FOUNDATIONS

OF ECONOMIC

THEORY

115

two can be run together, but the

third

raises

different

problems and

must be

kept separate.

With>(1) as an illustration, I would make first a mathematical

point

which

is

not

always

appreciated.

A

problem

of

maximum

(minimum)

subject to

single constraint

can

always

be

posed

in

two

ways with

identical results.

Generally,

the maximum

(minimum)

of

one function

is sought subject

to

the

constancy

of

a

second

function.

This can be

turned round

to give

the

minimum

(maximum)

of

the

second

function

subject

to a constant

value of the first. For

example,

let

x

=

X(VV2

.....x

. )

be the production

function

and

y

=

Y(VlV2 .

v. )

be the cost functionof a firm,the v'sbeing inputs.

Stage (1), as

posed above,

is

to

Minimize

y

for

given

x. The

solution

is

obtained

by minimizing

(y

-

Xx)

where

X is a

Lagrange

multiplier,

i.e.

dyax

-y=

X

ax(i

=

1, 2, ..............

)

avi

avi

with

x(v1, V2

.

v. A

)

=

x

=

constant.

This gives

y

and the v's

as

functions

of

x

(and

of the

given input

3y~

prices

which are The alternative formulation is to

maximize

avi)

x for

a given

y (maximum output at

a

given

cost.)

Here

we

maximize

(x

-

gy),

i.e.

ax

ay

(i

=

1, 2, .....

n)

avi

avi

with y

(VI,

2,

.

v

vn)

=

y

=

constant.

giving x and

the v's as functions of y (and

the

input

prices).

The

result is

identical with the

first, setting

=-X

and

inverting

x

as

a

function

of

y

into

y

as

a

function of x.

In

stages

(1) and (2)

together, with a

continuous

production

function,

one of the

few

meaningful

results which can be

obtained

is Vi

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